There are several reasons why hardening off is important, but cool night temperatures are toward the bottom of that list. At the top of the list is light intensity. Plants that need hardening off are, almost by definition, started indoors under lights. Most artificial light is relatively weak, and does not include the broad spectrum of wavelengths that are found in sunlight. Hardening off is a way to acclimate plants to that more intense light of the sun. If you are already putting your plants outside an a daily basis, you are achieving that goal. The second important effect of the hardening off process is structural; by exposing your plants to the effects of wind and rain, the cell structure in the stems and leaves is stressed, the tissue reacts to that stress by becoming more resistant to it, and the plant is less likely to suffer damage when moved from the protected environment to the outside.

Your plan, apparently, is to move the plants to a low tunnel. This is also a protected environment, and the plastic cover will in fact still act as a filter for some of the suns' rays, so in that situation hardening off is less crucial than if, as for many here, the process is to take indoor-grown seedlings and introduce them to an outdoor environment to mature. Your low tunnel will mitigate, slightly, the environmental effect that the plants might otherwise suffer from if transitioned, without hardening off, to an outdoor environment.

Lastly, the cool-season crops you list are relatively tolerant of the normal temperature variations that they will be experiencing in the early part of the season, so cool nights won't likely be a particular concern. Many plants, once established, can survive quite well under a blanket of snow, this is because plants suffer most, on a cellular level, not from cold, but from variations in temperature between freezing and thawing. Snow acts as an insulating blanket, keeping temperatures stable - this is the concept underlying the practice of applying winter mulches as well - and to a small degree the same effect will be created by your tunnels. The plastic will increase soil temperatures on sunny days, relative to conditions outside the tunnel, those temperatures will ameliorate the effects of radiational cooling at night, and the cooling effect of evaporation, increased by wind, won't be as much of an issue.

Perhaps the best next step in your hardening off process wold be to introduce a fan when the seedlings are indoors, and when they are outside, be sure to give them exposure to direct sunlight, not filtered even through plastic, whenever circumstances allow.

I agree that night time air temps, assuming it is above 45 degrees, are the least problem for your seedlings. It is intense sun exposure and/or harsh wind exposure that kill more than anything.

Early exposure to the breeze from a fan and to muted, dappled sun and shade is best. Direct sun for brief periods (1 hr or less) is ok but if they are to be left unattended then direct sun can quickly kill them.

Check out the FAQ here on Hardening Off.

Dave

Here is a link that might be useful: Hardening Off FAQ

I will definitely spend some time with the FAQ. The plants have been in the sunlight since they sprouted as I have not yet been able to get lights started adequately (hopefully by next year). I will begin working with the fan in the evenings on them. Wind will be less of a factor on these as when the plastic comes off the the tunnel, they will be under a row cover and my plan for these plants is to harvest all of them before that bed is later planted in summer squash(definitely before May 15th, which is my last average frost date).

If it helps to know, our daytime highs are in the 45s, and lows are about 26. Any snow cover protection is rather sporadic here.