Health Care Security Act of 2018 This bill amends the Internal Revenue Code to modify the rules for health savings accounts (HSAs) with respect to contribution limits, catch-up contributions for married couples, and medical expenses incurred before an HSA is established. The bill increases the maximum contribution limits for HSAs to equal the maximum for the sum of the annual deductible and out-of-pocket expenses that may be required to be paid for covered benefits under a high deductible health plan. If both spouses of a married couple have family coverage under a high deductible health plan, each spouse may make catch-up contributions to the same HSA. (Catch-up contributions are additional contributions which individuals who are at least 55 years of age may make to an HSA.) If an HSA is established within 60 days of the beginning of coverage under a high deductible health plan, any distribution from the HSA used to pay a qualified medical expense incurred during that 60-day period after the health coverage began is excludible from gross income. (Under current law, the medical expense must be incurred on or after the date that the HSA is established.)