Dale,
Are you tying the holidays to a specific year? For instance, Memorial Day, Labor Day and Easter change each year. Or are you just looking at specific holidays that stay the same every year? Just curious if you are going to simply put in the dates, or if you are going to calculate them out.
When it comes to dates, PHP date functions tend to convert to a seconds based timestamp (number of seconds since Jan 1st, 1970, I think). You can use time() to get a current snapshot, or strtotime() to set a date in time to a snapshot date.
So, convert all your holiday dates to snapshot dates within an array, convert your start and end dates to snapshot dates, and use a foreach() statement to evaluate each holiday date to see if it's between the start date and end date. If it is, increment a counter by 1. When done processing the foreach() loop, you have a count of how many holidays fit within the set.