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That would not be illegal.
It is illegal to discriminate against an employee based on membership in a legally protected class- race, color, national origin, gender, religion, age, disability. Unless you were able to tie the bed bugs to one of these classes, it would not be illegal to terminate the individual.
That said, you should be eligible for unemployment benefits, based on the reason for termination.
I am sorry, I know this is not what you hoped to hear, but this is the reality. My policy is to just give the truth, rather than tell you something false only for you to be more frustrated later when you discover the truth.