I'm not exactly SURE WHAT your position is because something is wrong with the details provided. My guess is that the strike prices are $60 and $90 and that this is a diagonal spread rather than what you are calling a calendar spread (aka horizontal spread) which is a long and short option position on the same underlying with the same strike price but with different expirations.
If that is correct then if AAPL rises and you are assigned on your short $90 call, $9,000 will be credited to your account and you will be short 100 shares at $90. It matters not what the current price of the stock is (the 5:30 price?) because the contractual obligation is to sell at $90. You will still be long the Jan 23, 2017 $60 call and the risk graph of this position is the same as that of the original diagonal spread since they are synthetic equivalents.
If this is standard Reg T 50% margin (brokers may require more than 50%) then initially you will need $4,500 of marginable securities and/or cash to support the position. Without getting too deep into the weeds, there is a minimum margin maintenance requirement and it will increase as the price of AAPL increases. However, since you own the $60 call, it will be increasing at the same rate and it will offset (standard options are not marginable but LEAPs are).
If you do not have the margin to support opening the short position then your broker will sell something of value to raise the cash to buy enough shares to cover and close the short position. One would hope and expect that your broker would close the long call but I have read enough horror stories on the web to know that you don't leave anything up to your broker.
Will your broker be upset with you? Possibly. Some brokers do not like having to intervene in client accounts. Whether this involves any subsequent restriction depends on the broker. The best thing to do is to contact your broker and find out how they handle such situations.