It makes no difference how long you've been with a company so yes, it's legal to lay off any and everyone without severance. The exceptions: a union agreement requiring severance, a personal contract calling for a severance. This is usually only for executives and ``key'' people.
No advance notice of termination or resignation is required. If advance notice of resignation is given, it can be accepted, rejected, or modified by the employer.
Texas employers are not required to offer severance payments to employees they lay off or terminate, unless there is a specific employment contract or company policy requiring the payments.
Under Texas law, good cause is the employee's failure to perform the duties that a person of ordinary prudence in the industry would perform under similar circumstances. However, because this definition is vague, parties should specifically state what constitutes good cause in their employment agreement.