If your board draws 600mA at 12V then the boost converter needs to output 12*0.6 = 7.2W. Assuming 80% efficiency the input power will be 9W, which is 2.4A at 3.7V. So the converter has to draw 4 times more current from the battery at 3.7V than it delivers to the load at 12V.
Peak switching current will be higher because a boost converter cannot run 100% duty cycle. At 90% duty cycle you would need 2.4/0.9 = 2.7A. The MC34063 can only switch 1.5A, so it is not suitable unless you add an external power transistor.
Typical Lipo cycle life is 100-500 cycles depending on current draw. However this is for 100% charge/discharge. Cell phones may limit charging and discharging to less than full capacity in order to extend lifespan. As the battery ages its internal resistance increases. This is more of a problem at high discharge rates, so a device that goes for several hours on a charge could get much longer lifespan than one which drains the battery in 10 minutes.
To calculate the battery capacity required you first need to know the average current draw. For example if the board draws 600mA 25% of the time and 200mA the rest of the time, the average current draw is 0.6*0.25+0.2*0.75 = 0.3A. Now translate that to battery current at 3.7V; 0.3*4 = 1.2A. To get 5 hours run time the battery would need to have a usable capacity of at least 1.2*5 = 6Ah (6000mAh).
If the average current is very low then a much smaller battery could be used, but the boost converter must also have low quiescent current draw. Most modern boost chips have very low Iq, but some older types are quite high. Manufacturer and supplier websites often have a parametric search function that you can use to select a suitable part, eg. Texas Instruments Step Up (boost) converters.
I don't know of any IC that charges a single lipo cell and boosts it to 12V. However there are several that boost to 5V. Is it possible that your board only uses 5V internally? If so then perhaps you could bypass its on-board regulator and power it with 5V.
Another option might be to use a 3S Lipo, which has a usable voltage range of 11~12V. Then you won't need a voltage booster. However you may want to include a method of checking and/or maintaining cell balance. Under/over voltage and over-current cutoff would also be a good idea. Protection circuit boards which do all that are readily available, or you could get a battery which has built in protection.