Which spyder version? Does it support dual display?
Secondly, if your laptop has only one graphics card (I think almost all only have one) then you cannot calibrate for two displays. The profile will be applied to both displays simultaneously. Finally only one display will be correct.
You could calibrate the eizo monitor and use this for the image, leaving the tool boxes on the laptop display. Alternatively, get an external (USB) graphics card.
Another point to note: not all web browsers recognize embedded ICC profiles. While PS displays everything correctly, Internet Explorer is known to ignore ICC profiles and render whatever it thinks is best. Firefox recognizes ICC profiles, Chrome as well (iirc).