Is it possible to unlock holiday achievements at other times of year?

You can override the active holiday in the launch options as if it were a different time of year:

1. Right-click Unturned in your Steam Library
2. Select Properties... to open the General tab
3. Find the Launch Options field
4. Type "-Holiday=XMAS" without quotes

Valid options other than XMAS are Halloween, HW, Christmas, PrideMonth, Valentines, and AprilFools.

Can I play with a different holiday in multiplayer?

In multiplayer the active holiday is controlled by the server. If you are hosting the server you can add the "-Holiday=XYZ" option to the server's command-line arguments to override the default.