Electrical – Low power switch with microcontroller

R1 and R2 will limit the current into pins on your uC and this is usually sufficient to protect your device - you just need to check in the spec what that "limit" current is and choose a resistor value that is appropriate given that the uC supply may be at 0V (un powered). The zeners can be left out on this basis.

Reliability is another issue. Switch bounce may cause your uC to switch on then switch off a few times so write your code to be aware of this.

I think it may be advisable to have a resistor on enable but probably in the region of +10K and maybe this could be higher possibly 100k.

The voltage on the shutdown pin has to be at least 45% of Vin so this shouldn't be an issue.

Some more info on the charger would be useful (any specs written on the casing, in the manual, etc)
Hopefully it's a switching charger.

Anyway, some rough calculations:

Battery Wh = 1.1Ah * 3.7V = ~4Wh Typical car battery Wh = 50Ah * 12V = 600Wh

You say the device operates for roughly 6 hours on a full charge, so the average current and power consumption is:

1.1Ah / 6 = 183mA 4Wh / 6 = 678mW

If we assume the battery is in good condition and can supply, say 300Wh (half of it's capacity) and still start (I'm no expert on car batteries so this is a bit if a guess, but I'm pretty sure less than 50% charge is a bad idea) then the operating time is:

300Wh / 0.678W = 442 hours or 18 days.

Now the above does not include the charger inefficiency and battery drain caused by other electronics in the car, so it seems feasible that you could halve this figure quite easily.

So although making sure the charger is doing a good job is certainly a good idea, I think it is likely that just shutting off the charger periodically may not be the answer, and you need to look at lowering the devices consumption (e.g. transmit data less frequently) and/or installing a larger capacity battery. Also, if the car is not in a garage, one of the readily available dashboard solar panels would help to keep the battery topped up.

Simple Timer Circuit

Since there has to be many such circuits already out there, rather than draw it from scratch I had a quick look around to find the kind of thing I had in mind that didn't involve a microcontroller (sorry it's a bit late, some urgent stuff came up)
Anyway, this circuit at http://www.electronics-project-design.com/electronictimerswitch.html seems to almost fit the requirements quite nicely, and is based on easily swappable and cheap components. It can be upgraded if necessary (e.g. the oscillator could be crystal based for better accuracy)

The IC is a 14-bit binary counter, and is clocked by the RC oscillstor made from pins 9, 10 and 11. The values of the RC components set the clock period (more info in the C4060B datasheet)
To set the on/off period, you need to AND the correct pins together for your desired ratio. Say you set the timing so the the count reaches 2^14 in 60 minutes, this means the clock is 2^14 / 3600 = ~4.55Hz.
Now you want the timer on for 7.5 minutes of this hour, so you need it on for 2^14 / 8 = 2048 counts. So to calculate the bits we need to AND together:

2^14 - (2^14/8) = 14336 to binary equals:

11100000000000

So bits 13, 12 and 11 need to be ANDed together (the same as the schematic above just without D1)
This is just a rough example, other timings can be achieved by calculating accordingly and you could use proper AND/OR gates, or a comparator IC or cascade another counter IC or... if you want to improve the design. Hope this helps to get you started.