I think the first kind will probably not live through your winters anyway. Otherwise

"Monitor your soil pH
closely and keep it between 5.6 and 7.0"

Here is a link that might be useful: Growing Raspberries & Blackberries in the Inland Northwest & Intermountain West PDF

We don't know what zone ftan lives in - Idaho has a wide range (zones 2 to 7), so I would not say that they will not live through the winter. It has not been below zero in over ten years here in Boise. The bulk of Idaho's population is in zone 6. Knowing where ftan lives would help.

Lowering the PH will likely help as the arid parts of Idaho (the parts that make up zones 5,6,7) do tend to have alkaline soil.

Here is a link that might be useful: Idaho Hardiness Zones

I've seen Boysenberry and related forms described as tender in local Extension publications. On the other hand, at least a few commercial web pages claim -10F.

Notice that it is not included in the table of recommended varieties at the pages I linked to above.

I planted Boysenberries years ago, and they failed to winter over in Northern Virginia. We rarely approach zero; never in the last 10 years that I can recall. I think that anything below 10 degrees will take out Boysenberries, or at least seriously damage them to the point they will not produce.

Don Yellman, Great Falls, VA

Boysen is one of the more cold-sensitive blackberry varieties. If raspberries have done well in the past without you having to manipulate the pH, I'm sure your Boysen will be fine.