Are there any tests that can predict whether I will have a premature menopause?

No, there aren’t any reliable tests for predicting the age that menopause will occur.
Female relatives of women who have undergone premature menopause are on average more likely to have an earlier menopause. In the very few instances in which a genetic mutation has been identified as the cause of premature ovarian insufficiency, female relatives can seek genetic counselling and be tested for the specific genetic mutation.
A blood test for AMH (or Anti-Müllerian Hormone) has been suggested to be able to predict when someone might enter the menopause. However, studies have produced very contradictory results regarding the usefulness of AMH for this purpose. At present, AMH is not recommended for predicting menopause onset. This year, the American College of Obstetricians & Gynaecologists (ACOG) concluded that “the use of antimüllerian hormone as a predictor of the onset of the menopause is unsuitable for clinical practice at this time”.
For more information on AMH, see my section on AMH, Ovarian Reserve and the Egg Timer Test.