Absolutely not the case, Dave - the share price could fall to zero but if the company is still trading and paying its bills as they fall due, it's solvent. If shares can't be traded (e.g. a private company rather than a quoted one) the shares are impossible to sell but that's no indication at all of the financial health of the company. Share prices for quoted companies will generally follow the fortunes of the underlying company, but they do not drive the company - it's the opposite way round.
Even if a company became insolvent, the shareholders would (should) get something back in the liquidation of the company's assets. That's an important baseline value to establish the minimum worth of a share.