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Starquad cables are generally used with the wires used in pairs, to reduce interference. They're often used for recabling headphones simply because they're a suitable replacement. You can buy starquad cables with four differently coloured inners, which will make your job easier. If you have a multimeter, that will make the job far easier. Take notes on what is connected where before you start desoldering, to stop yourself from getting in a muddle.
For the bridge cable between the two earpieces, you just need a two-core shielded cable of a reasonable size. The outer insulator may well be printed with the wire gauge, which will allow you to select a suitable replacement. Headphone cables don't carry much power and aren't particularly sensitive to impedance or capacitance, so I wouldn't worry too much about getting an exact match.