The European Union just won the Nobel Peace Prize, because the union has worked to prevent war on the continent for a long time. It is a good way for the countries to come together and work out problems. They are able to share solutions which makes them all stronger as a result.
The E.U. has similarities among it's individual members, that is, economic similarities, and if they stick together, I believe they will have firmer ground on which to stand when it comes to negotiating and also just plain validity as to how strong they are seen to be. If they separate, it'll be each entity for itself, and this can only weaken individual members, or strengthen just a few at the cost of hurting several others.
If the E.U. split up right now it would be an economic disaster far more widespread than the ones occurring/believed to be about to occur in the region. While working through one another may not be ideal of late, a separation would ruin virtually every nation involved. It isn't a matter of "should", right now they have to stick together.
At this point the E.U has to stick together. All the countries involved- maybe with the exception of the U.K- are so intertwined in each others economies and social polices that they can not dissolve. If a country like Germany or France were to leave many other countries would simply collapse. A collapse would lead to an economic crisis and simply a mess in Europe. The countries made a commitment to work together as one and now they need to live up to that.
From what I can gather, the arguments posted 'For' all focus on how this will benefit Europe, we live in Britain and should be focused on Britain and how this will benefit us. Case studies of the Swiss and Norway suggest that they benefit from virtually free access to the EU market with out the huge costs which are imposed on Britain. Each side has good argument to stay, but to stay to fund struggling EU countries and push our nation into further financial difficulty doesn't make sense. Under the EU agreement, the EU have to make a trade agreement with a member that leave, no one will know if this will be better or worse for Britain until it hits the negotiating table, at the moment both arguments are just speculation. At the moment Britain import more goods from Europe than it exports, so these trade agreements need to work both ways.