Initially, looking at all four digit primes consisting of middle number pairs (e.g. -00-, -11-, -22- ... -99-) indicates a range of 10 to 13 possibilities for any given pair, well beyond my limited computing/spreadsheet skills to fully analyse (I eagerly await Charlie's full spin on the solution).  One significant exception, however, is clearly 7 as an impossible value for 'x'.

There are only 6 such possibilities using -77- (i.e. 1777, 2777, 3779, 5779, 6779 and 8779).  Note that each ends with either a 7 or 9 (with no 1's or 3's).  That means each of the four squares in the bottom row must therefore consist of either a 7 or a 9 only.  There are, however, simply no prime numbers consisting of just (7 or 9) (7 or 9) (7 or 9) (7 or 9), arranged in any combination.  Ten primes therefore cannot be achieved with a grid having 7 as the value for 'x'.