This is a bit of a joke on the claim that homosexuality is a disease. If you had the flu, you might call your boss and ask for the day off - we would say that you "called in sick to work". Similarly, if you had some chronic health condition - say chronic back pain - you might occasionally 'call in sick' to work to ask for the day off because you're experiencing high levels of pain that day. So the idea here is that if homosexuality is a disease, you could calling in sick as with any other disease. To emphasize this, the word 'sick' is replaced with 'queer', since this is the "disease" you are calling work about.
"Call in queer to work", then, means to ask for the day off from work due to having the "disease" homosexuality.
I believe the idea here is to take the concept of homosexuality as a disease to an extreme, which then demonstrates more intuitively the inappropriateness of this conceptualization of homosexuality.