Just wondering if two channels are independent then is the overall capacity, the sum of the individual capacities?
Yes. That directly results from the very definition of independence and of capacity!
Now if these channels were independent and in a cascade would the overall capacity be 1−ε + 1−φ?
Oh! That's something different! You don't have two independent channels that you use, you have a single channel that is composed of a series of these two.
So, no, the sum can't be right. Think about this:
The first channel's mutual information is at most the original source's entropy (in fact, it's lower, as you notice, $1-\epsilon$). The second channel again has a mutual information that's at most as high as its input entropy, and that is already limited by the mutual information of the first channel.
Hint: try to make a probabilities "table" for outputs given inputs, and try to model this new combined channel to find out the true capacity. It's easy!