The mishna quotes a machlokes if a tziduki can be relinquish his rights in the courtyard - mevatel reshus (R' Gamliel) or if he's like a non-Jew and he can't be mevatel reshus (tanna kama). However, the gemara never seems to address the question if the tziduki can be part of the eruv in the first place. I think that there are three possibilities:
1. Everyone would agree that he can't be part of the eruv.
2. It would be the same machlokes - if you hold that he can't be mevatel then he can't be part of the eruv otherwise he could be part of the eruv.
3. Everyone would agree that he can be part of the eruv.
I think that Rashi would hold of option 1. In the mishna (on 61b) Rashi says that a tziduki can't be mevatel reshus because we're afraid he's going to carry there on purpose and thereby nullify his bitul. We don't have that fear if he's actually part of the eruv so that would be permissible. The Rambam (hilchos Eruvin 5:16) though seems to pasken like option 3. He paskens that a tziduki could be mevatel reshus but he can't be part of the eruv. I think that pashut pshat in the gemara is like option 2