Can both my husband and I claim property tax deductions on our home if we file married filing separate?

Yes, you can claim the amount of property taxes that you actually paid on separate returns, but the total on both returns should not equal more that the total paid on the property.

When you file married filing separately you both have to itemize or you both have to use standard deduction.   If you have chosen to itemized then you can use property tax.   The total amount entered between the two of you cannot exceed the total amount paid.  And....the deduction for each of you is capped at $5K, per federal tax law.