Moshe's wife was the daughter of a Midianite priest so she was not Jewish. Does this in fact mean Moshe's sons Gershom and Eliezer were not Jewish as well? There are various questions on Mi Yodeya regarding Moshe's offspring but I did not find anything regarding their being Jewish or not.
I'm inferring yes.
We see that God was angry that Moshe had not circumcised Gershom, and Tzipporah had to do it. While we know that technically, Gentiles have circumcised their children, there is no obligation to do so. The fact that God was angry implies that Moshe was obligated to do this meaning that Gershom (and, by further inference, Eliezer) were Jewish
The Zais Ranaan on the Mechilta shemos 18:2 says it's a machlokes tanaim whether Moshe converted his wife tzipora or not before marrying her. The one that holds he didn't convert her would hold Moshes sons were not Jewish. Most mefarshim indeed argue with the Zais Ranaan and hold everyone agrees Moshe converted his wife.