I am 60 years of age, and I am not diabetic. However I think I have low sugar. I have been feeling light headed dizzy and very shaky.
My friend loaned me her diabetic meter. I checked my blood sugar Sunday and it was 27 and Monday it was 43. I called my doctor and she is getting me my own meter. Should I be concerned, and if it is 27 again what should I do?

If you talked to your doctor about getting your own meter... why didn't you ask them when you were talking to them?

Whenever you feel shaky get something to eat. Peanut butter crackers and orange juice are recommended for low blood sugar.

And those readings are very low.

Yes. Those readings would indicate hospitalization. Normal readings are between 70 and 110. Readings that low could have caused a hypoglycemic coma.

Two common reasons for low readings are test strip not inserted all the way, and dehydration.
Since your symptoms are those of dehydration (among other possibilities), you should take care to drink enough liquids all day, especially in the summer heat.
And that doesn't include coffee, tea, and Coke!
I have a huge glass pub mug that I fill with tap water and a few drops of juice just to flavor it. Drink drink drink.
(I know I'm a hydration nut around AMHD, but that's necessary, I think. ERs are full of fainting people in the summer who have nothing else wrong with them.)