But the government could require such checks for bitcoin, since the cashier has to connect to the internet anyway in order to accept a bitcoin payment.
They can't actually do that, since either they are unsuccessful (much likely) or nobody would bother use this coin anymore, since its purpose would have been defeated (so they weren't able to control it, only to destroy it).
The stated purpose of bitcoin was to provide a method of internet payment that was fast cheap easy safe etc.. Some people thought that it was also a safe way to hide illegal payments from governments, and loved it for that perceived quality. Those people (and presumably most readers of this thread) are now realizing that it never was.
A tainted-bitcoins database would only kill
that use of bitcoin for good. Cashiers would surely comply and check that database if it was automatic, and/or if they could be penalized for failure to do so (if only by having their own coins listed, and hence made worthless).
Due to the mixing property of bitcoin transactions, black coins will slowly mix with white coins and the UTXO set will soon be filled with every shade of grey. Even the "virgin" coinbase output in each new block would become "tainted" by the transaction fees from any grey coins included in that block. Any attempt at blacklisting would then require drawing a line in the sand for how "grey" a coin could be before it should be refused. This line in the sand would likely be different across users, merchants and countries.
Markets would emerge that discount coins based on the coin's grey level. For example, if X is the greyness (1 being black and 0 being white), the discount would likely be linear in X:
effective BTC value = BTC value * (1 - k*X)
where k is the discount factor. And as soon as these markets are efficient, then users don't really need to care about how dark their coins are as long as the value the user sees is the value after discounting based on grayscale. Wallets could even deal with this in an automatic fashion as it would present an arbitrage opportunity. "Did someone send you bright white coins? Well, just blacken them down to the threshold and earn some free BTC!" The end result is effectively fungible money.
People of course realize this and the blacklist becomes a joke (or more likely, would never happen in any meaningful way in the first place).