Thanks for your question Dai.
The circumstances under which these buyouts can occur is based on the US takeover code and related legislation, as well as the laws of the state in which the company is incorporated.
It's not actually the case that a company such as Dell needs to entice or force every shareholder to sell. What is salient is the conditions under which the bidder can acquire a controlling interest in the target company and effect a merger. This usually involves acquiring at least a majority of the outstanding shares.
Methods of Acquisition
The quickest way for a company to be acquired is the "One Step" method. In this case, the bidder simply calls for a shareholder vote. If the shareholders approve the terms of the offer, the deal can go forward (excepting any legal or other impediments to the deal).
In the "Two-Step" method, which is the case with Dell, the bidder issues a "tender offer" which you mentioned, where the current shareholders can agree to sell their shares to the bidder, usually at a premium. If the bidder secures the acceptance of 90% of the shares, they can immediately go forward with what is called a "short form" merger, and can effect the merger without ever calling for a shareholder meeting or vote. Any stockholders that hold out and do not want to sell are "squeezed out" once the merger has been effected, but retain the right to redeem their outstanding shares at the valuation of the tender offer.
In the case you mentioned, if shareholders controlling 25% of the shares (not necessarily 25% of the shareholders) were to oppose the tender offer, there would be serveral alternatives. If the bidder did not have at least 51% of the shares secured, they would likely either increase the valuation of the tender offer, or choose to abandon the takeover. If the bidder had 51% or more of the shares secured, but not 90%, they could issue a proxy statement, call for a shareholder meeting and a vote to effect the merger. Or, they could increase the tender offer in order to try to secure 90% of the shares in order to effect the short form merger. If the bidder is able to secure even 51% of the shares, either through the proxy or by way of a controlling interest along with a consortium of other shareholders, they are able to effect the merger and squeeze out the remaining shareholders at the price of the tender offer (majority rules!).
Some states' laws specify additional circumstances under which the bidder can force the current shareholders to exchange their shares for cash or converted shares, but not Delaware, where Dell is incorporate.
There are also several special cases. With a "top-up" provision, if the company's board/management is in favor of the merger, they can simply issue more and more shares until the bidder has acquired 90% of the total outstanding shares needed for the "short form" merger. Top-up provisions are very common in cases of a tender offer.
If the board/management opposes the merger, this is considered a "hostile" takover, and they can effect "poison pill" measures which have the opposite effect of a "top-up" and dilute the bidders percent of outstanding shares. However, if the bidder can secure 51% of the shares, they can simply vote to replace the current board, who can then replace the current management, such that the new board and management will put into place whatever provisions are amenable to the bidder.
In the case of a short form merger or a vote to effect a merger, the shareholders who do not wish to sell have the right to sell at the tender price, or they can oppose the deal on legal grounds by arguing that the valuation of the tender offer is materially unfair. However, there are very few cases which I'm aware of where this type of challenge has been successful. However, they do not have the power to stop the merger, which has been agreed to by the majority of the shareholders.
This is similar to how when the president is elected, the minority voters can't stop the new president from being inaugurated, or how you can be affected if you own a condo and the condo owners' association votes to change the rules in a way you don't like. Tough luck for you if you don't like it!
If you want more detail, I'd recommend checking out a web guide from 2011 here as well as related articles from the Harvard Law blog here.
I hope that helps!