Sounds like you unchecked "Allow NULL value" for those fields in the form. Open the form in ArcPad Studio and look at the Control Properties for each field you want to have nulls.
If they were checked out from a geodatabase, "Allow NULL Values" was set to "no" on the fields. You will have to redo your database feature classes so that option is set to "yes" and redo your checkout. If you just make ArcPad allow NULL values, you face not being able to check in your edits if there are NULL values.
Shapefiles do not accept null values. See here for more information...
Hannah 🙂