What does a Culex mosquito do instead of blood feeding when it enters winter diapause?

Answer: It switches to sugary plant fluids and stores fat for the winter.

It switches to sugary plant fluids and stores fat for the winter. ✓ — That's the diapause strategy of Culex females. As days shorten, they stop blood feeding and instead seek carbohydrate sources like nectar and fruit juices, converting those sugars into fat reserves. The fat is what carries them through winter, and they shelter in protected places like culverts, caves, or unheated buildings until spring.

It lays winter-hardy eggs, and then the adults die off. — This describes the winter strategy of some other mosquitoes, such as Aedes albopictus, which survives as diapause eggs. Culex pipiens, however, overwinters as adult females, not eggs. So although eggs are a real overwintering route in the mosquito world, they are not what Culex does instead of blood feeding.

It develops thick, armor-like body parts to survive the cold. — Growing thicker body parts sounds like a reasonable cold-weather defense, but Culex doesn't rely on body armor to survive winter. Its adaptation is physiological and behavioral: it fuels up on sugars, stores fat, and finds a sheltered site. A thicker shell wouldn't provide the stored energy needed over months.