Darrin,
With careful planning, it should be possible to arrange your trip. Is there any possibility you could get 3 weeks for the trip?
You might consider something like this:
Flight to London (you'll lose the first day in flight times, normally arriving the day after you depart (flight times will be shorter from Boston of course - my flights from the west coast are usually 9-12H). If possible use LGW as it's much easier to navigate than LHR. Take the Gatwick Express into London Victoria station.
London - touring (a day trip to Bath?)
Train to Scotland (you could stop in York for one night if you're interested - it's a wonderful town with lots of history).
Scotland - I'd suggest Edinburgh over Glasgow, but your choice.
budget flight to Ireland (I haven't checked but I believe EasyJet has flights to Belfast - you might have to use RyanAir to Dublin)
Ireland - touring (you won't have a lot of time, so you'll have to be a bit "selective" in which places to visit. Some of my favorites are Kinsale, Galway and especially Dingle. Transportation in Irerland is not as "developed" as other parts of Europe, so you may have to use a combination of train and Bus to get to some locations.
Flight home - you might check flights from Shannon as well as Dublin.
Regarding accommodations, you might find that budget Hotels are in the same range as Hostels (for a double room). Some Hostels are co-ed, some are segregated by gender. The Guidebooks have listings in all the places you'll be. You could also check the HI Hostels website. These usually include breakfast (the HI in York had awesome breakfasts, but tell them to "hold" the Black Pudding!).
The end of May is a good time to travel, but it's in the spring shoulder season so it could be busy. Pre-booking accommodations and budget flights would be a good idea.
Be sure to pack light if taking budget flights, or you'll PAY!
Happy travels!