A constant means that it has the same value for every case.
For instance, if you include a variable for "taiwan" but there is nobody from taiwan (and it gets a 0 for everybody), this creates a problem, because you have no cases to estimate effects in Taiwan (i.e., they all have a 0). Thus, there is no variation in Taiwan.
I think something may have gone wrong when you created your dummies.
I did not really understand what you were saying when you said: "0= reference country, 1=country of birth".
Maybe take a look at this document, that explains how you can create dummy variables if you have more than 2 categories:
https://www.moresteam.com/whitepapers/d ... iables.pdf