The Outdoor Leadership Program has a sometimes confusing number of personal equipment requirements. Footwear for the program is at the top of the list. Understanding the need for water/portage shoes and camp shoes for the Raquette River expedition is one thing. Understanding the connection between boots for the backpacking trip and boots for the winter trip is a whole other issue!
No one shoe/boot is going to do it all. There is overlap between the types of conditions one might expect between the seasons and regions of the White Mountains, but the extremes it what we have to plan for. While the backpacking trip in mid-November could be quite warm and dry, it is better to plan for wet and winter-like at that time of the year and latitude.
Having a separate boot for the backpacking and one for the winter trip is one answer to accommodate the wide range of conditions expected. A fully waterproof and moderately insulated hiking boot with a gaiter is all that is necessary for back packing. Throw in a vapor barrier or neoprene sock and all is well for heavy conditions. For the winter expedition, a heavily insulated boot with a removable liner is required (pack boots, double plastics, etc.). Having both means being able to select the ideal choice for the given conditions. Having both means a great deal of expense as well.
It is possible to find one boot that will work for both trips, but may not be ideal for either. Finding a boot that can both hold up to putting in the miles and keeping you warm (and having a removable liners) can be a costly proposition. You many just have to be willing to suffer a bit. For example, using the winter boot on the backpacking trip when the weather turns warm can be a drag. Also, the winter boot may not be up to the rigors of pounding miles during the backpacking trip. Generally, we will not be putting in a great number of miles and many of those will be in wet areas off trail, so the physical structure (hike-ability) of the winter boot may not be an issue. If you are going to buy one boot for both trips you will need to buy for the worst-case conditions of the harshest season/location where they will be used.
Lots of research into the various properties of boots designed for different situations will undoubtedly need to happen. Being clear about the conditions you expect to encounter is the key. Always default to the worst possible scenario, particularly if you will be responsible for leading others. With the right boot you can free your mind to do the hard work of leading.