I disagree, Spirit. You are showing commitment to your nation by paying taxes to it, just like Jesus instructs. Since you pay taxes to it, you are accepting its authority over you & therefore swearing allegiance (loyalty, committment, accepting its superiority) to it. This is not blind allegiance, it's not binding, it's not saying it has more authority than God, it's not saying it's always right, it's simply accepting the present relationship with it. Just like any other relationship you may have.
You are a correct on a clarification I need to make on my last line: churches I've attended say the pledge after the service in a little optional memorial on very select occasions (holidays). I find nothing wrong w/ this. Do you vote? That's not directly in the Bible but I suspect it's something you do anyways. Do you read for pleasure? Do you watch movies? Do you listen to 'pagan' music? Not in the Bible, but you do it anyways. So yes, doing it in a service seems wrong, but in a church, with other believers, no, time & a place.
& no, there are times & places for things. When you're a little child, you don't do things an adult does, for example. It doesn't mean what the adult's do is wrong. It doesn't mean what the child's doing is wrong. 'There's a time & place' for things has always been understood as a truth.