If a butterfly is native in your area, then the chrysalis will generally have no problem surviving a normal winter. Because the chrysalis needs short days and cold temperatures to keep it in diapause, probably one of the best places you can keep it is outside in a protected area.
How cold can a monarch chrysalis survive?
First, the simple answer. For Monarchs and those that do not go into diapause as caterpillars, if the lows are above freezing and the day temperatures are above 65-70 F (18.33-21 C), they'll be fine. As cold-blooded critters, if temperatures drop too low, they'll literally freeze.
Do monarch caterpillars overwinter?
Each species winters in a particular stage. For example, swallowtails overwinter as pupae, mourning cloaks overwinter as adults, and viceroys overwinter as small larvae. Monarchs cannot tolerate the cold winters and must migrate to survive the winter temperatures.