All are correct, if only because the way that the adjectives work in this context is quite unusual.
The general adjective order in English is; Quantity or number, Quality or opinion, Size, Age, Shape, Color, Proper adjective (nationality, place of origin, material), Purpose or qualifier.
Now, "64-bit" could be considered quantity or size, but could also be considered purpose or qualifier, as could unsigned. As such, both positioning the size before or after the sign is justifiable.
Between "64-bit" and "64bit", the former is using a number, unit and hyphen to form an adjective, while the latter is combining a number and unit to give a measurement, which in turn is an adjective. Again, both are allowed.
Now, because we could consider the number to be a size, people will more often favour either 3 or 4 rather than 1 or 2. It's also more common to use the hyphen here, so 2 or 3 rather than 1 or 4. So while all four are allowed, "64-bit unsigned integer" would be the more common.
Plural forms would be incorrect, you don't use the plural when using a unit in a description, so "a 64-bit unsigned integer is an unsigned integer with 64 bits".