They could as blockchains can be a great tool for ensuring transparency and reliable data records. I think that this can be a great step forward for the government in terms of technology and I am sure that many countries would want to get ahead. However I can see that blockchain technology might also be not encouraged by corrupt governments. Imagine if they used the blockchain technology for elections, it would be near impossible for any politician to still cheat. This could be a reason why many would not be so into the technology because it will expose a lot of improper behavior from the government.
Aside from what I mentioned, this is technically and financially challenging. Not many countries would be so keen to spend lots of money for this technology especially if they do not see huge returns anyway.