Can consumer surplus be negative if a consumer is forced to make a purchase? Assume there is a good that a producer sells for $100$. A consumer's willingness to pay for that good is $50$. However a government program forces consumers to purchase the good irrespective of their willingness to pay (assume budget constraints are not an issue right now). Would this imply their consumer surplus would be $-50$?

Or do we reflect the negative consumer surplus as producer surplus (this doesn't make sense to me)?
Consumer surplus is their willingness to pay minus the price they pay,  and producer surplus is the price they receive minus their willingness to receive. So if you are assuming that consumers are forced to buy at a price of 100, yes the consumer surplus is negative. and according to your example, the producer surplus will be zero. You are right it does not make sense whatsoever to reflect it as producer surplus.