I have a "Real Easy" ear training book and it uses solfege, but I have a real hard time with that. I use a guitar tuner app to check myself and I find I can hum much better than singing "do's",ect. Is this going to be a major hindrance to my ear training or can I replace one with the other as I make my way through the book ?
The argument I heard as a student was that movable "do" allows you to make a mental association between scale degrees and solfege syllables. After a lot of practice singing with solfege syllables, the syllables tend to start popping into your head as you hear a melody. That gives you a way to fix those pitches in your mind and remember them, so that (for example), you can then transcribe the melody or play it on your instrument.
You asked: "...can I replace one with the other as I make my way through the book?"
I think the answer is no. Your book is using those syllables for good reasons; like Bruce wrote, they allow you to make a mental association among notes (whether your book uses movable "do" or fixed "do"). As you progress further in your book, I imagine those syllables will become more and more critical to the training contained within it.
Singing in syllables is not an intuitive skill. It takes a great deal of practice in the beginning. Stick with it, and get good at it. There are huge benefits to being able to sing by syllables! It may seem strange at first, but it will make sense later, so drink the "coolaid," so to speak, and have faith.
When I went to school for music, my professor suggested we all use musictheory.net. They have a bunch of different excercises that help with ear training. I suggest atleast looking at it.