You are not wrong, you need a confocal microscope that is equipped with an spectrum detection system or unmixing in case the signals overlap. In the regular confocal you can provide the emission and excitation, but typically for these two fluorophores the separation is not easy, but possible.
If possible, try to use a white laser that allows you the excitation of a single specific wave lenght... If not possible, set the conditions in the confocal of the EYFP so that you do not recover any signal when exciting with the red laser, then, any signal when using the Alexa556 should be your antibody... could you try Alexa 594?
Effectively, I used a Leica TCS SP5 equipped with white laser and eliminating the crosstalk between two fluorophores. Selecting the profile in both spectrum and subtract them if I have emission in both. But for some reason to a referee don´t like the selection of that combination, unfortunately the experiment was done in that condition. I suppose that i must to explain in a better way the analysis conditions.