My family follows the rule not to eat fish and chicken on Tuesday, Thursday, and Saturday.
Last Saturday, when I was ordering take-out from a restaurant, the person in front of me ordered chicken biriyani, but asked the chef to not put any chicken pieces in the biriyani, because he was observing the same rule as me.
Bengali Hindus refrain from eating non-vegetarian food on Thursdays because they believe Thursday is sacred to Bhagavatî Lakshmi. Since Tuesday & Saturday are considered to be sacred to Bhagavatî Dakshinâkâlî by Shâkta Âgamas(vide definition of Kula days of the week by Prânatoshanî Tantra), Bengali Hindus are free to eat non-vegetarian food on those days. I have also met people who refrain from eating non-vegetarian food on Mondays(not on Thursdays) because Monday is sacred to their family deity, Bhagavâna Shankara.
Hindus from Hindi-speaking regions refrain from eating non-vegetarian food on Tuesdays & Saturdays because those days of the week are sacred to Hanumâna.
Besides it is common misconception that non-vegetarian food is tâmasika.
Translation: Tâmasika people like the food which is stale, tasteless, putrid & rotten, refuse & impure.
Âdi Shankara's commentary on this verse:
Stale: Half cooked;
Yâtayâma(lit. cooked 3 hours ago), meaning 'powerless' is thus explained, to avoid tautology, for the next word 'gatarasa'(tasteless) means the same i.e 'powerless'; Rotten: Cooked food over which one night has passed; Refuse: Left after a meal; Impure: Unfit for offering
Out of all kinds of animals; only goat, lamb & buffalo meat can be offered at rituals in present conditions. So duly sanctified meat can't be considered Tâmasika.
Avoiding non-veg food as a part of Dharma is a resultant of how we classify different types of foods.
foods are distinguished as saatvik, Rajasi and taamsik. Without going too much in detail on types of food, I want to highlight that non-veg food generally high in tam which means that it takes one away from the spiritual path (if consumed without justification and only for taste).
When you don't eat non-veg on Thursday per-se, you make sure that you are on your way to following Vaishnav way of living which means you are(Probably a few inches only :)) closer to moksha. This does not justify eating it on other days, but you try to give respect and pay your regards to that God by keeping yourself less indulged in tamsik pravriti.
Like any library, Hinduism Stack Exchange shares great information, but does not offer personalized advice, and does not take the place of seeking such advice from any Acharya, Pundit, astrologer, Guru or other trustworthy Counselor.