What thermal paste and how to apply it to stop Probook 6550 overheating?

You are correct. The machine will shut down or Blue screen if it over heats. The over heating can be caused by the Heatsink on the CPU or GPU not providing a good enough heat conducting channel for cooling. Cleaning the fan is a good start, but also be aware of where the PC stands, is there sufficient ventilation for example to extract the hot air?
Being an All -in-One will make more difficult. But check if all electrical connections are secure as the power supply can also cause your machine to shut down and the power supply fan should also be cleaned.
To apply Thermal paste you have to dismantle the machine and get to the Motherboard. I suggest you take it to a repair centre as the dismantling can be difficult in these units.

Usually the notebooks shut down when increase the temperature of cpu. For thst reason we must open and clean the heating sink and apply new thermal paste on cpu and graphics card gpu chip every 2 with 3 years.

strip the laptop down, get to the cooling fan for the CPU and GPU and remove it. loosen the screws in accordance with the numbers on the sink. remove existing thermal paste with nail varnish remover and a clean cloth off both fan and CPU and apply fresh stuff. this is important because the old paste is burnt on and no longer aiding heat to pass through it. also the laptop is shutting down after a while because the fan heatsink is clogged with debris clean that out by removing the little fan itself out of the heatsink. this is most likely your problem if the laptop is over 2 years old. any help you need just ask :)

Three causes of ovheating:
1) Dust. Dust inside the computer, Dust around the vents, fans, cpu and heat sink. Take a can of compressed air and clean the dust out of the computer.

2) A Fan not working. Make sure the fan inside the Power Supply where you plug the power cord in works. Also there is a fan that sits on top of the CPU and Heat Sink. Make sure it works. Normally the system will give you an error message if the fan is not working.

3) Thermal Paste - The Thermal Paste around the cpu and heat sink has dried up or is too thin to keep the unit cool. Apply a fresh coat of Thermal Paste on the CPU and Heat Sink.

This may be an overheating issue. When a computer gets hot it has a fail safe to shut off to protect the system from perminate damage.

you might want to replace the CPU fan and re apply a thermal paste between your processor and heatsink. Make sure the heatsink and CPU fan is securely in place to avoid thermal problems

Listen for fans coming on, as this sounds like overheating issue. If fans kick in and then stay constant or dont work at all.
The link is for Notebook hardware control, this will monitor the temp among other things (only reference as a guide any software can give you an indication but not exact temp) Generally if a laptop hits 90c it will auto cutout to stop major hardware failure normal running temp (40-55c upwards depending on use) http://www.pbus-167.com/
If the machine is overheating it could be due to blocked vents, faulty heatsink, fans not working or incorrect fitting and thermal paste.
In all honesty if this is due to overheat problems, as the machine has been cutting out my strong advice would be to do all the above ESPECIALLY re-apply thermal paste (arctic silver5) to cpu, it takes about 100-200 hours of use to start to work 100%.
hope this helps, let me know if i can be of further help

The laptop is shutting down because the CPU is overheating (thermal shutdown) and if this problem is not fixed the CPU will eventually fail.
Replace the faulty fan and remove the dust from the heatsink.
While you are at it I suggest you apply fresh thermal paste on the CPU and hearsink.

This is typical of a overheating (thermal shut down) CPU problem.
When the CPU overheats and shuts down, it won't bootup again until the CPU cools down. If this is not fixed then the CPU will fail.
This problem is caused by either a faulty fan or the fan and heat assembly is clogged with dust.
Check the fan to see if it spins freely, replace if faulty and clean the dust off the fan and heatsink assembly.
Also apply fresh thermal paste to the heatsink/CPU.

no...u have a problem in hardware and is not working as good so try this 1st trouble(1) remove the memory then make a clean(2) remove the mini HDDrive then put it back again(3) if their a same again a problems,last step (4) you need to make a re-programs or re-install the Windows/OS...Good Luck and Have a nice Day..^_^