Whoever told you that the government cannot own a business is, quite simply wrong. There is no legal prohibition against the government owning a business.
There would not be a conflict of interest. However, at least arguably, there is an unfair advantage to that business. If the government owns a business, it can charge lower prices than the competition (with any deficit funded by the taxpayer). It also gets other advantage (like infusions of cash when necessary).
But these are all policy issues. There is nothing in the Constitution, or the law, that prohibits the government from owning a business. Perhaps it is a bad idea, but not against any law.
As for those who say that the US Government does not own GM, they are ... well ... wrong. Anybody who owns a share in a corporation is a part owner. Generally, a person with even 20 percent of a large, publicly traded company has effective control over it. In this case, the US government has limited voice on the Board of Directors. But, owning 60 percent of the stock, the U.S. Government has the RIGHT to call a meeting of shareholders and, since they own more than 50 percent of the shares, could name the entire board of directors, who, in turn, elect the officers of the corporation. The United States has a "controlling interest" in General Motors.
It is true that the government hopes to sell its shares. When it does, it will no longer own that portion of the corporation that it curently does. But, like any shareholder, it has the right to vote at shareholder meetings, etc.
Good luck.