I think the big thing to ask is whether any obligations or liabilities will be put on you as a shareholder. Probably nothing to worry about but you need to ask it. The response you get should be documented so you have evidence.

You will need to sign a shareholders agreement. So read it and try to understand it. Shareholders should get some form of annual reporting around the finances of the business. Sometime there are shareholder consents/votes required around particular events like issuing new stock, dilution, big loans, etc.

MAke sure you have pre-emption rights. This means that if another shareholder wants to sell shares he must offer them to existing shareholders first before selling to an outside party. This is normal and at least gives you an option to protect yourself from dilution. The same principle should apply to investor financing via convertible loan notes or indeed any form of debt.

Are you being asked to pay for your shares? If so then be alot more cautious with regards to signing anything. If you are getting shares for free this could be considered a gift and may be subject to gift tax. If gift tax is an issue make sure you are getting shares for as low a value as possible. Since your company has laid people off there is a case it is losing money and so has little value. Of course the existing investor may be very much against the idea of giving away shares a low price since it greatly devalues their investment and could be used by new investors down the road as evidence that the company had little value at one stage so how can you value it so highly now.

Are you being offered a directorship? A directorship comes with more duties and responsibilities but you will attend board meetings, get access to financial accounts and have an influence on how company is run. You can also develop a relationship with the major investors in the company who most likely are non-executive directors too.

Another thing is warranties and indemnities. When a company is sold shareholders can be asked to give warranties and indemnities against contingent events for a period of time (usually several years). This means if some tax liability came up from when you were a shareholder or ongoing court case went against the company then the old shareholders would be asked to cover the shortfall. Maybe you can exclude yourself from this now. Of course if you become an executive manager with shares then it would be perfectly normal for your you to give warranties and indemnities. The only type of shareholder who would have a case to escape the W&Is would non executive or employee shareholders as they can argue they were not the lead decision makers in the business.

Also there can be good and bad leaver terms tied to your shares. This could mean that if you left the company you could be asked to return your shares. Avoid this if possible. Even if you leave the company it would be nice to hold on to the shares. You never know what will happen in the future.

Finally make sure you get share certs and that the public company filings accurately reflect your share position. Some small companies are very bad at handling red tape and doing filings.

This area is complex and there's no perfect answer so I've probably missed a load of things you should watch out for. I am not a professionally qualified accountant or lawyer so take my advice at completely your own risk!