Do I have a case if I am the only employee to get a pay cut and my employer lied about it?

Unfortunately, you probably do not have any case, unless you have an employment contract restricting or limiting the company's right to fire you or cut your pay. (It doesn't have to be a personal contract; a union or collective bargaining contract would do as well.) The problem is, companies may fire people at will as long as their is no contract (and no discrimination against a protected category--no racial, gender, age, etc. discrmination). They don't have to be fair, and they don't even have to tell  the truth--the company has a right to cut your pay and your pay only, if they choose, and they can be deceitful in what they say about it.

If there is a contract, it'd be a very different outcome--they'd have to honor the contract's terms.

I know that this situation seems unfair but it is probably legal.  Most employment relationships are what is known as "at will".  This means that basically the employer can hire or fire someone for any reason or no reason whatsoever, as well has increase/decrease salary/hours, promote/demote, and generally impose requirements as they see fit.  In turn, you can work for an employer, or not, your choice. 

The only exceptions to the above would be if there is a stated company policy covering this, or there is a union/employment agreement that governs, or this situation has arisen due to some type of discrimination.