Heating would definitely be an issue but because you want to grow cutting flowers, strong lights and sufficient growing space will be really important. I keep a lot of things growing and dormant in my unheated, windowless garage using supplemental light and heat, but your project would be on another level
Here in zone 6 with coldest winter lows down to negative single digits, my garage can freeze down to mid-20's, requiring extraordinary measures to keep DORMANT borderline growing zone plants alive.
First, I think what you need is not so much a greenhouse with clear enclosure intended to allow sunlight in, but an insulated smaller space inside the garage (don't forget floor and ceiling). I would use reflective material for walls and ceiling -- maybe even straight insulation boards with mylar sheeting -- to reflect all of the supplemental lighting (and heat) back inside. A grow room. But you maybe able to get away with just florescent shoplights if you choose the right kind of plants.
For choice of plants, I am not good with cutting flowers, but I'm thinking you may need to think outside the box since typical inexpensive cutting flowers might be annuals, and reblooming perennials would go dormant. If you were to start annual flowering plants from seeds now, even if successful, I don't think they will grow to bloom until winter or even spring.
I wonder what kind of flowering plants you have in the garden right now that might be divided before the ground freezes? I wonder if any of them can be forced to grow and bloom? Hmm... zone 4 -- it may be too late for you this year. You may need to start gathering plant materials earlier in the season.
Some of the typical grocery store flowering gift plants might be a good reference... You might even use them as a source for this year.
I think what you want are plants that bloom in cooler season -- mums and asters that are blooming now could be kept blooming for a while. I wonder if some of those endless summer hydrangea gift plants could be kept blooming?
From the fall garden, perennial ageratum, snakeroot, goldenrod could have been dug up while still blooming. If you were to keep them pruned DURING their growth in the summer, they would be more manageable size to dig up in late summer. I think various flowering mints makes great addition to cut flowers.
You could also consider colorful foliage plants like coleus and ferns with interesting leaf structures. Maiden Fern for example. Oh and not really fern but asparagus fern is an ubiquitous cut bouquet green. So that would mean a lot of the foliage HOUSE plants -- especially variegated ones -- could make good additions too.
Force-able flowering bulbs might be something that could be started now. Are strongly scented flowers a no-no in the hospital? Paper Whites are easy, and there might be others like Freesia. I don't know which flowering bulbs make good cut flowers.
Hanging basket of vining plants and fuchsias might be another option.
You may also want to get into orchid growing -- at least I have a very long gorgeous orchid show of just phalaenopsis and hybrids from my 4 plants in late fall/early winter through spring. But I do keep these in the house at upper 60's. Cut moth orchid flowers last for a long time floated in a bowl. Those might be for very special patients or a common area. If you are successful in growing the kind of orchids with large sprays of small flowers, I imagine they could be made into small sprigs.
Later on, in late winter/early spring, one of the ways I enjoy cut flowers is to cut budding spring flowering branches from trees and shrubs outside -- apple, cherry, plum ...hmmm can't think of a shrub right now... and force them in the house. If you have suitable trees for this, you could do the same.
Wow I already wrote a lot, but a couple more outside-the-box ideas -- many plants will grow better in cold air with warmer roots, so I would spend the money for horticultural heating mats and soil heating cable if you grow in a larger container. SIP (self irrigated planter) could be kept warmer with aquarium heater in the supply reservoir or use livestock waterer heater or heated dog water bowl. For that matter, a large tubs of water on the floor kept warm (50's-60's) with the water heater could provide extra warmth and can be used for watering the plants with. You can use siphoned tubing to link multiple tubs.
If you keep a vermicomposter and a couple of bokashi fermenter in the space, they will generate heat.
the ability to tap into the brains of others! Great info! After some thought, and all the great suggestions, I'm thinking of forcing some bulbs for sure; also looking at deadheading mums and keeping them going, perhaps try rose. I think more likely at this point is area in my basement. Of course I'll still need to use my growlights. But, although it's not fully heated, we have a special heater that keeps temps above 60 at least- during the coldest of periods. I love the idea of the Mylar/reflective material. Makes great sense. And I'm afraid my skills are WAY too elementary for orchids 
. Thanks again