Passover in Gilgal
This page presents the rationale for considering it as a Pinpointed Lunar Sabbath, and discusses several objections.
Joshua 5:10-12 describes the Children of Israel keeping Passover in the land of Gilgal:
And the children of Israel encamped in Gilgal, and kept the Passover on the fourteenth day of the month at even in the plains of Jericho. And they did eat of the old corn of the land on the morrow after the passover, unleavened cakes, and parched corn in the selfsame day. (the 16th). And the manna ceased on the morrow after they had eaten of the old corn of the land; neither (on the same morrow) had the children of Israel manna any more; but they did eat the fruit of the land of Canaan that year. (Jos 5:10-12)
Joshua and the Children of Israel precisely followed the instructions of Leviticus 23:10-14:
Speak unto the children of Israel, and say unto them, When ye be come into the land which I give unto you, and shall reap the harvest thereof, then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before YHVH, to be accepted for you: on the morrow after the sabbath the priest shall wave it. And ye shall offer that day when ye wave the sheaf an he lamb without blemish of the first year for a burnt offering unto YHVH. And the meat offering thereof shall be two tenth deals of fine flour mingled with oil, an offering made by fire unto YHVH for a sweet savour: and the drink offering thereof shall be of wine, the fourth part of an hin. And ye shall eat neither bread, nor parched corn, nor green ears, until the selfsame day that ye have brought an offering unto your God: it shall be a statute for ever throughout your generations in all your dwellings. (Lev 23:10-14)
On the 14th of Abib the Children of Israel kept the Passover at even (i.e. at Sunset, beginning the 15th of Abib), then ate unleavened bread and parched corn on the morrow of the Passover (the 15th) which would be the 16th. Neither bread or grain are allowed to be eaten until the Wave Sheaf Offering is presented. Since the Wave Sheaf Offering is supposed to be offered the day after the Sabbath, and they ate bread and grain on the 16th, this proves that the 15th had to be a Sabbath.
There are several objections regarding whether the 15th of the month was, in this case, a Sabbath.
Morrow of the Passover
It is assumed in the rationale above that the "morrow after the Passover" refers to the 16th. However, this exact same phrase "morrow after the Passover" (or ממחרת הפסח in Hebrew) also appears in Numbers 33:3
And they departed from Rameses in the first month, on the fifteenth day of the first month; on the morrow after the passover the children of Israel went out with an high hand in the sight of all the Egyptians. (Numbers 33:3)
This passage makes it clear that the "morrow after the Passover" refers to the 15th day of the month and not the 16th. Therefore the day that the grain of the land was eaten was the 15th, and so the Sabbath must have been on the 14th.
Allowable Timing of the Sabbath
It's widely believed that the Sabbath must have been on the 15th, and then the grain eaten on the 16th, because the Sabbath in question regarding the Wave Sheaf Offering should not occur before the Days of Unleavened Bread. However, this requirement is not found anywhere in scripture! Note that in the instructions for the Wave Sheaf Offering, the subject at hand is the day AFTER the Sabbath, not the Sabbath itself:
Speak unto the children of Israel, and say unto them, When ye be come into the land which I give unto you, and shall reap the harvest thereof, then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the LORD, to be accepted for you: on the morrow after the sabbath the priest shall wave it. (Lev 23:10-11)
If it's more important for the Wave Sheaf Offering to occur during the Days of Unleavened Bread than the "Sabbath" in question to do so, then the Wave Sheaf Offering could have been offered on the 15th as the plain reading of the text of Jos 5:11 implies... which would mean that the 14th was a Sabbath and not the 15th.
Criteria for the Wave Sheaf Offering
It can be argued that the Wave Sheaf Offering did not actually occur at this time. It is widely assumed that the Wave Sheaf Offering took place between Jos 5:10 and 5:11, since the Passover was celebrated in v10 and then the Israelites ate of the grain of the land in v11. However, we need to look closely at the requirements of the Wave Sheaf Offering to see if it would have been possible for them to observe it at this time in question.
Three times thou shalt keep a feast unto me in the year. Thou shalt keep the feast of unleavened bread: (thou shalt eat unleavened bread seven days, as I commanded thee, in the time appointed of the month Abib; for in it thou camest out from Egypt: and none shall appear before me empty:) And the feast of harvest, the firstfruits of thy labours, which thou hast sown in the field: and the feast of ingathering, which is in the end of the year, when thou hast gathered in thy labours out of the field. (Exo 23:14-16)
Exodus 23:16 specifies that the Feast of Harvest (i.e., the Wave Sheaf Offering) was to be offered from grain produced "of thy labors" and "which thou hast sown in the field". If the Israelites had just arrived in the land, and were eating grain that had been planted by the peoples which they had driven out, then a sacrifice of that grain would not meet this requirement. Furthermore, Leviticus 22:25 also specifies that an offering "from a stranger's hand" is not acceptable:
Ye shall not offer unto the LORD that which is bruised, or crushed, or broken, or cut; neither shall ye make any offering thereof in your land. Neither from a stranger's hand shall ye offer the bread of your God of any of these; because their corruption is in them, and blemishes be in them: they shall not be accepted for you.(Lev 22:24-25)
Therefore it is possible that the Wave Sheaf Offering was not offered at this time, due to the fact that the Israelites had not planted this grain themselves but had taken it from foreigners. This would then mean that the Sabbath could have been any day that week.