Gas discharge tubes contain elemental mercury in rather small quantities, and the health risk is very minimal. Elemental mercury is not readily absorbed through the skin or GI tract, and it would require the ingestion of orders of magnitude higher quantities than is present in neon tubes/fluorescent tubes to cause any real problems. The main hazard of elemental mercury is toxicity via inhalation of vapors, but this requires a much higher vapor concentration than is possible with a single (or even several) broken tubes in a volume of air the size of a typical den or game room.

Remember that mercury amalgam dental fillings (approximately 50% Hg, 50% Ag/Sn/Cu alloy) have been used safely for at least 100 years.

It's only with mercury compounds that toxicity (and explosion risk!) starts getting scary, especially organic mercury compounds and water-soluble inorganic salts.

If a tube/light bulb breaks, sweep (don't vacuum) up the debris, seal it in a freezer bag, then dispose of it in a CFL recycling bin (Lowes, Home Depot, and Menards have them).

If you are really worried, keep a mercury spill kit on-hand: https://www.amazon.com/ADC-Mercury-Spill-Kit-980SK/dp/B000F96DHA