Generally, no — in Jainism, spinach and amaranthus are usually allowed as vegetables.

What Jains definitely avoid are mainly:

• root vegetables like onion, garlic, potato, carrot, beet, radish, etc.
• vegetables that involve more violence in harvesting in some strict observances

Spinach is a leafy vegetable, so it is not normally banned in Jain diet. Amaranthus:

• as a leafy vegetable is generally allowed
• as rajgira/amaranth grain is also commonly eaten by Jains

That said, some very strict followers may avoid certain greens on specific fasting days or during special vows.

So the simple answer is: Jains usually do eat spinach and amaranthus.