The primary loyalty of each and every human being is to the G-d Who created him. Loyalty to G-d comes before all other loyalties. And since the Jewish people have been historically a Theocratic government-in-exile in which the Rabbinate had civil authority (the idea of Jews as a "religious denomination" is a modern one) insistence that Jews place loyalty to another country before what G-d has commanded of them is very tricky.
Secular states made up of various religions are a relatively recent phenomenon in history. At any rate, to place country before G-d is idolatry. Too many "G-d, family, country" conservatives imply that G-d is a projection or totem of the family/country rather than an objective Truth.
Fundamentalism and conservatism do not always mesh (contrary to popular belief). I am most definitely a religious fundamentalist, which means I object to G-d sublimated to any other loyalty or loyalty to G-d being made use of for another reason.
I don't want to get into a religious debate but I never think about loyalty and God at the same time. I see loyalty as something that can be shifted (if done correctly and I'm not going to go into detail here). Whereas my relationship with God is not about loyalty it just is.